I'm not much of a DQ player myself, so I can't help you there.
I am however a solo Q climber. So I can shed some light I believe.
- Farm. Farming is the most important part of league from the 1-12 minute mark by far. Focus on farming and dodge or avoiding your opponents harass. If you can trade, do so. But do not go out of your way to harass in most instances. Keep in mind that a kill is worth about 15-20 farm, so if you get a huge lead in farm, you can afford to give up a kill.
- Objectives. It's simple, and I'm sure you know. But Dragon, and Herald are seriously powerful and very important. I'm sure you know to get turrets so I won't go into that.
- Learn how to freeze lane and farm safely under turret. If you're playing someone the opponent cannot trade with, freezing the lane and denying farm is going to severely tilt your opponent, and may even cause them to leave.
- Learn to end. When you score an ace or take down 3-4 of their champions, find the best location the push, and shove the hell out of those turrets.
These are just some of the basics, but they should be enough to get you out of silver no problem. If you want to get out of gold later, I'll introduce a few more mechanics you'll have to learn. Good luck climbing!